We report on the measurement of spin density matrix elements of the Lambda(1520) in the photoproduction reaction gamma p -> Lambda(1520)K+, via its subsequent decay to K- p. The measurement was performed as part of the GlueX experimental program in Hall D at Jefferson Laboratory using a linearly polarized photon beam with E-gamma = 8.2 GeV-8.8 GeV. These are the first such measurements in this photon energy range. Results are presented in bins of momentum transfer squared, -(t - t(0)). We compare the results with a Reggeon exchange model and determine that natural exchange amplitudes are dominant in Lambda(1520) photoproduction.